General Relativistic Evolution Equations for Density Perturbations in Closed, Flat and Open FLRW Universes

P. G. Miedema∗ Netherlands Defence Academy Hogeschoollaan 2 NL-4818CR Breda The Netherlands March 9, 2021

Abstract It is shown that the decomposition theorems of York, Stewart and Walker for symmetric spatial second-rank tensors, such as the perturbed metric tensor and perturbed Ricci tensor, and the spatial fluid velocity vector imply that, for open, flat or closed Friedmann-Lemaître-Robertson-Walker universes, there are exactly two, unique, independent gauge-invariant quantities which describe the true, physical perturbations to the energy density and particle number density. Using these two new quantities, evolution equations for cosmological density perturbations and for entropy perturbations, adapted to non-barotropic equations of state for the pressure, are derived. Density perturbations evolve adiabatically if and only if the particle number density does not contribute to the pressure. Local density perturbations do not affect the global expansion of the universe. The new perturbation theory has an exact non-relativistic limit in a non-static universe. The gauge problem of cosmology has thus been solved.

pacs: 04.25.Nx; 98.80.-k; 98.80.Jk keywords: Cosmology; Perturbation theory; mathematical and relativistic aspects of cosmology arXiv:1410.0211v4 [gr-qc] 7 Mar 2021

∗[email protected]

1 Contents

1 Introduction 2 1.1 Gauge Problem of Cosmology ...... 3 1.2 Previous Attempts to solve the Gauge Problem of Cosmology ...... 3 1.3 Solution of the Gauge Problem of Cosmology ...... 4

2 Preliminaries 5 2.1 Choosing a System of Reference ...... 5 2.2 Equations of State for the Pressure ...... 6

3 Einstein Equations and Conservation laws for Closed, Flat and Open FLRW Universes6 3.1 Background Equations ...... 6 3.2 Linearized Equations ...... 7

4 Decomposition of the Spatial Metric Tensor, the Spatial Ricci Tensor and the Spatial Fluid Velocity8 4.1 Tensor Perturbations ...... 8 4.2 Vector Perturbations ...... 9 4.3 Scalar Perturbations ...... 9

5 Linearized Equations for Scalar Perturbations in Closed, Flat and Open FLRW Universes9

6 Unique Gauge-invariant Quantities 11

7 Non-relativistic Limit 12

8 Evolution Equations for the Density Contrast Functions 15

9 Thermodynamics 17 9.1 Pressure and Temperature Perturbations ...... 17 9.2 Diabatic Density Perturbations ...... 18 9.3 Adiabatic Density Perturbations ...... 18

10 Conclusion 19

A Derivation of the Evolution Equations using Computer Algebra 20 A.1 Derivation of the Evolution Equation for the Energy Density Perturbation ...... 20 A.2 Derivation of the Evolution Equation for the Entropy Perturbation ...... 22 A.3 Evolution Equations for the Contrast Functions ...... 23

1 Introduction

The theory related to the linearized Einstein equations and conservation laws is important in cosmology because it describes the growth of all kinds of structures in the expanding universe, such as stars, galaxies and microwave background fluctuations. Lifshitz(1946) and Lifshitz and Khalatnikov(1963) were the first researchers to derive a cosmological perturbation theory. They encountered the problem that the solutions of the linearized Einstein equations and conservation laws have no physical significance. Due to

2 the linearity of the equations, the solutions can be changed by a linear coordinate transformation, i.e., a gauge transformation. Therefore, the solutions are gauge-dependent. This is the well-known gauge problem of cosmology which is up till now not solved.

1.1 Gauge Problem of Cosmology Consider a closed, flat or open Friedmann-Lemaître-Robertson-Walker (flrw) universe filled with a perfect fluid, with an equation of state for the pressure which depends, according to thermodynamics, on the particle number density n and the energy density ε. The evolution of the energy density ε(0) and the particle number density n(0) of the background universe are governed by the usual Einstein equations and conservation laws for a flrw universe. The general solution of the linearized Einstein equations and conservation laws contains, among other quantities, ε(1) and n(1). Since the linearized equations are invariant under linear coordinate transformations xµ → x0µ = xµ − ξµ, one can generate new, equivalent, 0 0 solutions ε(1) and n(1), i.e., 0 0 0 0 ε(1) = ε(1) + ξ ε˙(0), n(1) = n(1) + ξ n˙ (0), (1) 0 0 where ξ ε˙(0) and ξ n˙ (0) are gauge modes. Therefore, ε(1) and n(1) are gauge-dependent and have, as a consequence, no physical significance. In contrast to ε(1) and n(1) the real, measurable density perturbations phys phys ε(1) and n(1) are independent of the choice of a coordinate system, i.e., are gauge-invariant. The physics of density perturbations is hidden in the general solution of the linearized Einstein equa- phys phys tions and conservation laws. Therefore, the physical quantities ε(1) and n(1) can be expressed as linear combinations, with time-dependent coefficients, of gauge-dependent solutions of these equations, such that phys phys the gauge modes are eliminated. Furthermore, ε(1) and n(1) have the property that in the non-relativistic phys phys limit the linearized Einstein equations and conservation laws combined with expressions for ε(1) and n(1) phys reduce, in a non-static flat flrw universe, to the time-independent Poisson equation with ε(1) as source phys phys 2 term, and the well-known Einstein relation ε(1) = n(1) mc . Consequently, the gauge problem is, in fact, phys phys the problem of finding expressions for ε(1) and n(1) .

1.2 Previous Attempts to solve the Gauge Problem of Cosmology Bardeen(1980) was the first to demonstrate that using gauge-invariant quantities, one can recast the linearized Einstein equations and conservation laws into a new set of evolution equations that are free of non-physical gauge modes. Bardeen’s definition of a gauge-invariant density perturbation is such that it becomes equal to the gauge-dependent density perturbation in the limit of small scales. The underlying assumption is that on small scales the gauge modes vanish so that gauge-dependent quantities become phys phys gauge-independent. As will be shown in Section7, the physical quantities ε(1) and n(1) do not become equal to the gauge-dependent quantities ε(1) and n(1) in the non-relativistic limit. Furthermore, the gauge modes εˆ(1) and nˆ(1) never become zero, since the universe is not static in the non-relativistic limit. Finally, the relativistic space-time gauge transformations reduce in the non-relativistic limit to Newtonian gauge transformations with space and time decoupled. Consequently, the quantities defined by Bardeen are not equal to the real physical energy density perturbation. The article of Bardeen has inspired the pioneering works of Ellis and Bruni(1989); Ellis et al.(1989); Ellis and van Elst(1998), Mukhanov et al.(1992); Mukhanov(2005) and Bruni et al.(1992). These researchers proposed alternative perturbation theories using gauge-invariant quantities which differ from the ones used by Bardeen. The theory of Mukhanov et al.(1992) yields the Poisson equation of the Newtonian Theory of Gravity only if the Hubble function, i.e., the expansion scalar, vanishes. This, however, violates the background Friedmann equation, as will be shown in Section7. In fact, the background Friedmann equation is needed to derive the Poisson equation from the linearized Friedmann equation in the non- relativistic limit. Consequently, the gauge-invariant quantities defined by Mukhanov et al.(1992) are not equal to the true physical perturbations. In the approach of Ellis and Bruni(1989); Ellis et al.(1989); Ellis and van Elst(1998) density perturbations are defined using gradients. They define a gauge-invariant and covariant function which ‘closely

3 corresponds to the intention of the usual gauge-dependent density contrast function’. However, a perturbation theory based on this deﬁnition does not yield the Poisson equation in the non-relativistic limit. The above mentioned pioneering treatments caused an abundance of other important works, e.g., Malik and Wands(2009); Knobel(2012); Peter(2013); Malik and Matravers(2013); Kodama and Sasaki (1984); Ma and Bertschinger(1995). The review article of Ellis(2017) discusses the work of Lifshitz and Khalatnikov, and also provides an overview of other research on the subject. From the vast literature on cosmological density perturbations, it must be concluded that there is still no agreement on which gauge-invariant quantities are the true energy density perturbation and particle number density perturbation. None of the cosmological perturbation theories in the literature has a correct non-relativistic limit as explained in Section7, so that there is still no solution to the gauge problem of cosmology.

1.3 Solution of the Gauge Problem of Cosmology phys phys As said before, the true, physical, density perturbations ε(1) and n(1) are hidden in the general solution of the linearized Einstein equations and conservation laws and it has proved difficult to determine in advance which linear combination of gauge-dependent quantities are equal to these true density perturbations. To solve this problem, the decomposition theorems of York, jr.(1974), Stewart and Walker(1974) and Stewart(1990) for symmetric three-tensors are used. In the literature, these decomposition theorems are only applied to the perturbed metric three-tensor. This, however, is not sufficient. The decomposition of the perturbed Ricci three-tensor must also be taken into account. Only then the linearized momentum constraint equations are consistent with the decomposition of the fluid three-velocity into a rotational part and a divergence part. This makes it possible to decompose the linearized equations into three independent systems which govern the evolution of scalar perturbations, vector perturbations and tensor perturbations. Only the part of the perturbed Ricci three-scalar which is associated with scalar perturbations is non-zero, so that only scalar perturbations are coupled to ε(1) and n(1). This fact, which is explained in detail in Section4, is essential since for scalar perturbations the three linearized momentum constraint equations can now be rewritten as one first-order ordinary differential equation for the perturbed Ricci three-scalar. This, in turn, implies that, for scalar perturbations, one exclusively needs the constraint equations and conservation laws. The fact that these equations are the perturbed counterparts of the background equations is crucial. It is well-known that the evolution of the unperturbed universe is determined by three independent scalars, namely the energy density, the particle number density and the expansion scalar, see Section 3.1. It has now been achieved that these three scalars occur also in the linearized equations in Section5, this time as perturbations. In Section6 it is shown that this substantially reduces the number of gauge-invariant quantities which can be written as linear combinations of gauge-dependent solutions of the linearized equations. As a consequence, there are exactly three non-trivial gauge-invariant quanti- gi gi gi gi phys ties, namely, ε(1), n(1) and θ(1) = 0. In the non-relativistic limit, Section7, it is found that ε(1) ≡ ε(1) , gi phys gi phys gi phys gi phys gi phys n(1) ≡ n(1) and θ(1) ≡ θ(1) = 0. Therefore, ε(1) ≡ ε(1) , n(1) ≡ n(1) and θ(1) ≡ θ(1) = 0 holds true also in the General Theory of Relativity. phys phys Finally, evolution equations for ε(1) and n(1) are derived using the algorithm given in the appendix, phys whereas θ(1) = 0 implies that local density perturbations do not affect the global expansion θ(0) of the universe. In fact, the cosmological perturbation theory in Section8 consists of a second-order ordinary differential equation—with source term entropy perturbations—which describes the evolution of relative perturbations in the total energy density, and a first-order ordinary differential equation that describes the evolution of entropy perturbations. The coefficients of the second-order equation depend on the equation of state p(n, ε) for the pressure and is, as a consequence, different from the equations found in the literature. The first-order differential equation for the entropy perturbation is a new result. With the evolution equations for density perturbations given in Section8, the gauge problem of cosmology for closed, flat and open flrw universes has been solved.

4 2 Preliminaries

2.1 Choosing a System of Reference phys phys In order to derive evolution equations for ε(1) and n(1) a suitable system of reference must be chosen. Due to the general covariance of the Einstein equations and conservation laws, the General Theory of Relativity is invariant under a general coordinate transformation xµ → x0µ(xν ), implying that there are no preferred coordinate systems (see Weinberg(2008), Appendix B). In particular, the linearized Einstein equations and conservation laws are invariant under a general linear coordinate transformation, i.e., a gauge transformation xµ → x0µ = xµ − ξµ(t, x), (2) where ξµ(t, x) are four arbitrary inﬁnitesimal functions of time, x0 = ct, and space, x = (x1, x2, x3), coordinates, the so-called gauge functions. Since there are no preferred systems of reference and since phys phys the solutions ε(1) and n(1) of the evolution equations are invariant under the inﬁnitesimal coordinate transformation (2), i.e., are gauge-invariant, one may use any coordinate system to derive the evolution equations. A suitable system of reference can be chosen by the following two considerations. Firstly, in order to gi gi unambiguously identify the gauge-invariant quantities ε(1) and n(1) found in Section6 as the real density phys phys perturbations ε(1) and n(1) , one needs the non-relativistic limit. In the Newtonian Theory of Gravity space and time are strictly separated, implying that in this theory all coordinate systems are essentially synchronous. In view of the non-relativistic limit it is, therefore, convenient to use synchronous coordinates (Lifshitz, 1946; Lifshitz and Khalatnikov, 1963; Landau and Lifshitz, 2010) in the background as well as in the perturbed universe. In these coordinates the metric tensor gµν (t, x) of flrw universes has the form

2 g00 = 1, g0i = 0, gij = −a (t)˜gij(x), (3) where a(t) is the scale factor of the universe, g00 = 1 indicates that coordinate time is equal to proper 1 time, g0i = 0 is the global synchronicity condition (see Landau and Lifshitz(2010), § 84) and g˜ij is the metric tensor of the three-dimensional maximally symmetric sub-spaces of constant time. From the Killing µ equations ξµ;ν + ξν;µ = 0 and (3) it follows that the functions ξ (t, x) in (2) become

∂ψ(x) Z ct dτ ξ0 = ψ(x), ξi =g ˜ik(x) + χi(x), (4) ∂xk a2(τ) if only transformations between synchronous coordinates are allowed. The four functions ψ(x) and χi(x) cannot be ﬁxed since the four coordinate conditions g00 = 1 and g0i = 0 have already exhausted all four degrees of freedom, see Weinberg(1972), Section 7.4 on coordinate conditions. Secondly, a property of synchronous systems of reference is that the space-space components of the four-dimensional Ricci curvature tensor Rµν are split into two parts such that one part contains exclusively all time-derivatives of the space-space components of the metric tensor gij and the second part Rij is the Ricci curvature tensor of the three-dimensional sub-spaces. In fact, Rij is the three-dimensional Ricci tensor which is expressed in terms of gij in the same way as Rµν is expressed in terms of gµν . All operations of raising indices and of covariant diﬀerentiation are carried out with the three-dimensional metric gij, Landau and Lifshitz(2010), § 97. These properties make it possible to apply the decomposition theorems of York, jr.(1974), Stewart and Walker(1974) and Stewart(1990) to the perturbed metric three-tensor and perturbed Ricci three-tensor. In conclusion, a synchronous system of reference is the most appropriate coordinate system to derive phys phys the evolution equations for the density perturbations ε(1) and n(1) .

1Quantities with a tilde, i.e., q˜, belong to the three-dimensional maximally symmetric subspaces of constant time.

5 2.2 Equations of State for the Pressure Barotropic equations of state p = p(ε) do not take into account the particle number density, so that the inﬂuence of entropy perturbations on the evolution of density perturbations cannot be studied, see Section 9.3. Since heat exchange of a perturbation with its environment could be important for structure formation in the early universe, realistic equations of state are needed. From thermodynamics it is known that both the energy density ε and the pressure p depend on the particle number density n and the absolute temperature T , i.e., ε = ε(n, T ), p = p(n, T ). (5) Since T can, in principle, be eliminated from these equations of state, a computationally more convenient equation of state for the pressure is used, namely

p = p(n, ε). (6)

This equation of state for the pressure and its ﬁrst- and second-order partial derivatives have been included phys phys in the evolution equations for the density perturbations ε(1) and n(1) .

3 Einstein Equations and Conservation laws for Closed, Flat and Open FLRW Universes

The background and linearized equations can be calculated from the set of Einstein equations (97.11)– µν (97.13) from the textbook of Landau and Lifshitz(2010) and the conservation laws T ;ν = 0.

3.1 Background Equations The complete set of background Einstein equations and conservation laws for open, flat and closed flrw universes filled with a perfect fluid with energy-momentum tensor

T µν = (ε + p)uµuν − pgµν , p = p(n, ε), (7) is, in synchronous coordinates, given by

2 1 4 3H = 2 R(0) + κε(0) + Λ, κ = 8πGN/c , (8a) ˙ R(0) = −2HR(0), (8b)

ε˙(0) = −3Hε(0)(1 + w), w := p(0)/ε(0), (8c)

ϑ(0) = 0, (8d)

n˙ (0) = −3Hn(0). (8e)

The G0i constraint equations and the Gij, i 6= j, dynamical equations are identically satisfied. The Gii dynamical equations are equivalent to the time-derivative of the G00 constraint equation, or Friedmann equation (8a). Therefore, the Gij dynamical equations need not be taken into account. In equations (8) Λ is the cosmological constant, GN the gravitational constant of the Newtonian Theory of Gravity and c the speed of light. An over-dot denotes differentiation with respect to ct and the sub-index (0) refers to the background, i.e., unperturbed, quantities. Furthermore, H :=a/a ˙ is the Hubble function which is, for 1 µ flrw universes, equal to H = 3 θ(0), where θ(0) is the background value of the expansion scalar θ := u ;µ µ −1 µ µ with u := c U the fluid four-velocity, normalised to unity (u uµ = 1). A semicolon denotes covariant differentiation with respect to the background metric tensor g(0)µν . The spatial parts of the background i i Riemann tensor R(0)jkl, the Ricci curvature tensor R(0)j and its contraction R(0) are given by 2K 6K Ri = R˜i = K δi g˜ − δi g˜ ,Ri = − δi ,R = − ,K = −1, 0, +1, (9) (0)jkl jkl k jl l jk (0)j a2 j (0) a2

6 where R(0) is the global spatial curvature. The quantity ϑ(0) is the three-divergence of the spatial part of µ µ µ the ﬂuid four-velocity u(0). For an isotropic universe the ﬂuid four-velocity is u(0) = δ 0, implying that

ϑ(0) = 0, so that there is no local ﬂuid ﬂow. From the system (8) one can conclude that the evolution of an unperturbed flrw universe is determined by exactly three independent scalars, namely

µν µ µ ε = T uµuν , n = N uµ, θ = u ;µ, (10) where N µ := nuµ is the cosmological particle current four-vector, which satisﬁes the particle number µ conservation law N ;µ = 0,(8e), see Weinberg(2008), Appendix B. As will become clear in Sections6 and7, the quantities (10) and their perturbed counterparts play a key role in the evolution of cosmological density perturbations.

3.2 Linearized Equations The complete set of linearized Einstein equations and conservation laws for open, ﬂat and closed flrw universes is, in synchronous coordinates, given by

˙ k 1 Hh k + 2 R(1) = −κε(1), (11a) ˙ k ˙ k h k|i − h i|k = 2κ(ε(0) + p(0))u(1)i, (11b) ¨i ˙ i i ˙ k i i h j + 3Hh j + δ jHh k + 2R(1)j = −κδ j(ε(1) − p(1)), (11c)

ε˙(1) + 3H(ε(1) + p(1)) + (ε(0) + p(0))θ(1) = 0, (11d) 1 d h i (ε + p )ui − gik p + 5H(ε + p )ui = 0, (11e) c dt (0) (0) (1) (0) (1)|k (0) (0) (1)

n˙ (1) + 3Hn(1) + n(0)θ(1) = 0, (11f)

µν µν i where hµν := −g(1)µν and h = +g(1) with h00 = 0 and h0i = 0 is the perturbed metric tensor, h j = ik ij g(0)hkj, and g(0) is the unperturbed background metric tensor (3) for an open, flat and closed flrw universe. Quantities with a sub-index (1) are the perturbed counterparts of the background quantities with a sub- index (0). A vertical bar denotes covariant differentiation with respect to the background metric tensor 0 0 g(0)ij. Note that the G(1)0 and G(1)i constraint equations (11a) and (11b) contain at most first-order time- i ˙ i derivatives of the metric. They relate the initial conditions h j(t0, x), h j(t0, x), ε(1)(t0, x), n(1)(t0, x) and i u(1)(t0, x). Once the constraint equations are satisfied at t = t0, they are satisfied automatically at all times. This is an intrinsic property of the Einstein equations (Weinberg(1972) and Landau and Lifshitz (2010), § 95), which will be used in Section5. The perturbation to the pressure is given by the perturbed equation of state for the pressure

∂p ∂p p(1) = pnn(1) + pεε(1), pn := , pε := , (12) ∂n ε ∂ε n where pn(n, ε) and pε(n, ε) are the partial derivatives of the equation of state p(n, ε). Using Lifshitz’ formula, see Lifshitz and Khalatnikov(1963), equation (I.3) and Weinberg(1972), equation (10.9.1), k 1 kl Γ(1)ij = − 2 g(0)(hli|j + hlj|i − hij|l), (13) and the contracted Palatini identities, see Lifshitz and Khalatnikov(1963), equation (I.5) and Weinberg (1972), equation (10.9.2), k k R(1)ij = Γ(1)ij|k − Γ(1)ik|j, (14) the perturbed Ricci three-tensor can be written as

1 kl R(1)ij = − 2 g(0)(hli|j|k + hlj|i|k − hij|l|k − hlk|i|j). (15)

7 Raising the index i, one gets, using also (9), i : ik ik 1 i 1 il k k k 1 kl i 1 i R(1)j = (g Rkj)(1) = g(0)R(1)kj + 3 R(0)h j = − 2 g(0)(h l|j|k +h j|l|k −h k|l|j)+ 2 g(0)h j|k|l + 3 R(0)h j. (16) ij k ij k Using that g(0)h i|j|k = g(0)h i|k|j, one ﬁnds for the contraction of the perturbed Ricci three-tensor, i.e., the perturbed Ricci three-scalar, : k ij k k 1 k R(1) = R(1)k = g(0)(h k|i|j − h i|k|j) + 3 R(0)h k. (17)

Expression (17) is the local perturbation to the global spatial curvature R(0) due to a local density perturbation. µ µ µ Finally, θ(1) is the perturbation to the expansion scalar θ := u ;µ. Using that u(0) = δ 0, one gets 1 ˙ k : k θ(1) = ϑ(1) − 2 h k, ϑ(1) = u(1)|k, (18) µ where ϑ(1) is the divergence of the spatial part of the perturbed ﬂuid four-velocity u(1).

4 Decomposition of the Spatial Metric Tensor, the Spatial Ricci Tensor and the Spatial Fluid Velocity

York, jr.(1974), Stewart and Walker(1974) and Stewart(1990) showed that any symmetric second-rank three-tensor, and hence the perturbation tensor hij and the perturbed Ricci tensor R(1)ij, can uniquely be decomposed into three parts. For the perturbed metric tensor, one has

i i i i k k k h j = hkj + h⊥j + h∗j, h⊥k = 0, h∗k = 0, h∗i|k = 0. (19)

The perturbed Ricci tensor, R(1)ij, being a symmetric second-rank three-tensor, can in the same way be decomposed

i i i i k k k R(1)j = R(1)kj + R(1)⊥j + R(1)∗j,R(1)⊥k = 0,R(1)∗k = 0,R(1)∗i|k = 0. (20) i Finally, York and Stewart demonstrated that the components hkj can be written in terms of two independent potentials φ(t, x) and ζ(t, x), namely 2 hi = (φδi + ζ|i ). (21) kj c2 j |j As will be shown in the next three subsections, the decompositions (19) and (20) and the expression (21) 1 2 3 imply that the decomposition of the spatial part u(1) = (u(1), u(1), u(1)) of the perturbed ﬂuid four-velocity

u(1) = u(1)k + u(1)⊥, (22) where the components u(1)k and u(1)⊥ have the properties ˜ ˜ ˜ ˜ ∇ · u(1) = ∇ · u(1)k, ∇ × u(1) = ∇ × u(1)⊥, (23) is consistent with the momentum constraint equations (11b). In expressions (23) ∇˜ is the generalised ˜ k k vector differential operator, defined by ∇iv := v |i. Using the decompositions (19), (20) and (22) the system of equations (11) is split into three independent systems. The solutions of these systems are referred to as tensor perturbations ∗, vector perturbations ⊥ and scalar perturbations k. These perturbations are briefly discussed in the following three subsections.

4.1 Tensor Perturbations

Using (19) and (20), equations (11a)–(11c) imply that ε(1) = 0, p(1) = 0 and u(1) = 0, so that tensor perturbations are not coupled to density perturbations. With (18) it follows that θ(1) = 0. This, in turn, implies that equation (8e) and (11f) are identical, so that n(1) = 0. As a consequence, the system (11) reduces for tensor perturbations to ¨i ˙ i i h∗j + 3Hh∗j + 2R(1)∗j = 0. (24) Due to the form of this equation, tensor perturbations are usually referred to as gravitational waves.

8 4.2 Vector Perturbations k From (11a), the contraction of (11c), R(1)⊥k = 0 and (19) it follows that ε(1) = 0 and p(1) = 0. Raising the ij k ˜k index i of equations (11b) with g(0), one ﬁnds, using also that Γ(0)ij = Γ ij, ˙ kj kj j h⊥|k + 2Hh⊥|k = 2κ(ε(0) + p(0))u(1), (25)

ij ij k ij where it is used that g˙(0) = −2Hg(0). It follows from R(1)⊥k = 0,(17) and (19) that h⊥ must obey

kl h⊥|k|l = 0, (26)

k in addition to h⊥k = 0. Taking the covariant derivative of (25) with respect to the index j and using (26) one ﬁnds that equations (25) reduce to uj = 0, or, equivalently, ∇˜ · u = 0. This implies with (18) that (1)|j (1)

θ(1) = 0, so that n(1) = 0. With (23) it follows that u(1)⊥ is coupled to vector perturbations. For vector perturbations, equations (11) reduce to the set of equations

˙ k h⊥i|k + 2κ(ε(0) + p(0))u(1)⊥i = 0, (27a) ¨i ˙ i i h⊥j + 3Hh⊥j + 2R(1)⊥j = 0, (27b) 1 d h i (ε + p )ui + 5H(ε + p )ui = 0. (27c) c dt (0) (0) (1)⊥ (0) (0) (1)⊥ In view of (23), vector perturbations are also called rotational perturbations.

4.3 Scalar Perturbations k Since both hkk 6= 0 and R(1)k 6= 0 scalar perturbations are coupled to ε(1), n(1) and p(1). It is now demonstrated that u(1)k is coupled to scalar perturbations, by showing that equations (11b) require that the rotation of u(1) vanishes, if the metric tensor is of the form (21). Diﬀerentiating (11b) covariantly with respect to the index j and subsequently substituting expression (21) yields

˙ ˙|k ˙|k 2 2φ|i|j + ζ |k|i|j − ζ |i|k|j = κc (ε(0) + p(0))u(1)i|j. (28)

Interchanging i and j and subtracting the result from (28) one gets

˙|k ˙|k 2 ζ |i|k|j − ζ |j|k|i = −κc (ε(0) + p(0))(u(1)i|j − u(1)j|i), (29)

˙ ˙ ˙|k ˙|k where it is used that φ|i|j = φ|j|i and ζ |k|i|j = ζ |k|j|i. By rearranging the covariant derivatives, equation (29) can be cast in the form

˙|k ˙|k ˙|k ˙|k ˙|k ˙|k 2 (ζ |i|k|j − ζ |i|j|k) − (ζ |j|k|i − ζ |j|i|k) + (ζ |i|j − ζ |j|i)|k = −κc (ε(0) + p(0))(u(1)i|j − u(1)j|i). (30)

Using the expressions for the commutator of second-order covariant derivatives (see Weinberg(1972), Chapter 6, Section 5)

i i i k k i i i k i A j|p|q − A j|q|p = A kR(0)jpq − A jR(0)kpq,B |p|q − B |q|p = B R(0)kpq, (31) and substituting the background Riemann tensor (9) for the three-spaces of constant time one ﬁnds that ˜ the left-hand sides of equations (30) vanish identically, implying that ∇ × u(1) = 0. Therefore, only u(1)k is coupled to scalar perturbations.

5 Linearized Equations for Scalar Perturbations in Closed, Flat and Open FLRW Universes

phys phys The true density perturbations ε(1) and n(1) are part of the scalar perturbations. Therefore, a new set of equations will be derived which describe exclusively scalar perturbations. To that end, equations (11) will be rewritten using the results of Section4.

9 i i i Since scalar perturbations are only coupled to hkj and u(1)k, one may replace in (11)–(18) h j by i i i hkj and u(1) by u(1)k, to obtain perturbation equations which exclusively describe the evolution of scalar perturbations. From now on, only scalar perturbations are considered, and the subscript k is omitted. Using the decompositions (19), (20), (22) and (23), it will be shown that the perturbed Einstein equations and conservation laws (11) can for scalar perturbations be written in the form

1 2H(θ(1) − ϑ(1)) = 2 R(1) + κε(1), (32a) ˙ 2 R(1) = −2HR(1) + 2κε(0)(1 + w)ϑ(1) − 3 R(0)(θ(1) − ϑ(1)), (32b)

ε˙(1) = −3H(ε(1) + p(1)) − ε(0)(1 + w)θ(1), (32c) 1 ∇˜ 2p p˙ ˙ 2 (1) 2 : (0) ϑ(1) = −H(2 − 3β )ϑ(1) − 2 , β = , (32d) ε(0)(1 + w) a ε˙(0)

n˙ (1) = −3Hn(1) − n(0)θ(1). (32e)

The system (32) consists of the energy density constraint equation or linearized Friedmann equation (32a), the momentum constraint equation (32b), the energy density conservation law (32c), the momentum conservation law (32d) and the particle number density conservation law (32e). As will be shown below, for scalar perturbations the dynamical equations (11c) are automatically fulfilled by the solution of the system (32). This is an intrinsic property of the Einstein equations and conservation laws. Note the remarkable similarity between the systems (8) and (32). In fact, the system (32) is the perturbed counterpart of the system (8). Just as in the unperturbed case (8), the evolution equations (32) consist of one algebraic equation (32a) and four first-order ordinary differential equations (32b)–(32e) for the five unknown quantities θ(1), R(1), ε(1), ϑ(1) and n(1).

Using that p˙(0) = pnn˙ (0) + pεε˙(0) and the conservation laws (8c) and (8e) the quantity β(t) becomes

2 n(0)pn β = pε + , (33) ε(0)(1 + w)

2 where pε and pn are given by (12). The symbol ∇˜ denotes the generalised Laplace operator with respect ˜ 2 ij to the three-space metric tensor g˜ij, deﬁned by ∇ f :=g ˜ f|i|j. The derivation of the evolution equations (32) for scalar perturbations is now given. Firstly, eliminating ˙ k h k from (11a) with the help of (18) yields the algebraic equation (32a). ij Secondly, multiplying both sides of equations (11b) by g(0) and taking the covariant derivative with respect to the index j, one ﬁnds

ij ˙ k ˙ k g(0)(h k|i|j − h i|k|j) = 2κ(ε(0) + p(0))ϑ(1), (34) where also (18) has been used. The left-hand side of (34) will turn up as a part of the time-derivative of ij ij the curvature R(1). In fact, differentiating (17) with respect to time, one gets, using also g˙(0) = −2Hg(0) and (8b), ˙ ij ˙ k ˙ k 1 ˙ k R(1) = −2HR(1) + g(0)(h k|i|j − h i|k|j) + 3 R(0)h k, (35) where it is used that the operations of taking the time-derivative and the covariant derivative commute, k ˜k since the background connection coefficients Γ(0)ij = Γ ij are independent of time for flrw metrics (3). ˙ k 0 Combining (34) and (35) and using (18) to eliminate h k yields (32b). Thus, the G(1)i momentum constraint equations (11b) have been recast in one first-order ordinary differential equation (32b) for the perturbed Ricci three-scalar, i.e., the local spatial curvature due to a density perturbation. Thirdly, it is shown that the dynamical equations (11c) are not needed. Clearly, the off-diagonal (i 6= j) equations (11c) are not coupled to scalar perturbations, so that these equations need not be taken into account. For the diagonal (i = j) equations, it is sufficient to consider the contraction of (11c). Eliminating ˙ k the quantity h k with the help of (18), one arrives at

˙ ˙ 3 θ(1) − ϑ(1) + 6H(θ(1) − ϑ(1)) − R(1) = 2 κ(ε(1) − p(1)). (36)

10 Using (32a) to eliminate the second term of (36) yields for the contraction of (11c)

˙ ˙ 1 3 θ(1) − ϑ(1) + 2 R(1) = − 2 κ(ε(1) + p(1)). (37) This equation is identical to the time-derivative of the constraint equation (32a). Diﬀerentiation of equation (32a) with respect to time yields

˙ ˙ ˙ 1 ˙ 2H(θ(1) − ϑ(1)) + 2H(θ(1) − ϑ(1)) = 2 R(1) + κε˙(1). (38) ˙ ˙ Eliminating the time-derivatives H, R(1) and ε˙(1) with the help of (8a)–(8c), (32b) and (32c), respectively, yields the dynamical equation (37). Consequently, for scalar perturbations the dynamical equations (11c) are not needed.

Finally, taking the covariant derivative of (11e) with respect to the background metric tensor g(0)ij and using (18), one gets 1 d h i (ε + p )ϑ − gik p + 5H(ε + p )ϑ = 0, (39) c dt (0) (0) (1) (0) (1)|k|i (0) (0) (1) where it is used that the operations of taking the time-derivative and the covariant derivative commute. ˜ 2 ij With (3), ∇ f :=g ˜ f|i|j and (8c) one can rewrite (39) in the form

˜ 2 ˙ p˙(0) 1 ∇ p(1) ϑ(1) + H 2 − 3 ϑ(1) + 2 = 0. (40) ε˙(0) ε(0) + p(0) a

2 Using the deﬁnitions w := p(0)/ε(0) and β :=p ˙(0)/ε˙(0) one arrives at equation (32d). The decomposition theorems of York, Stewart, and Walker have been used to rewrite the full system of Einstein equations and conservation laws (11) into a substantially simpler system of equations (32) which describes exclusively scalar perturbations. With the derivation of equations (32) the ﬁrst step towards the solution of the gauge problem of cosmology has been taken. In the next section, the background equations (8) and the linearized equations (32) phys phys will be used to identify which gauge-invariant quantities could be equal to ε(1) and n(1) .

6 Unique Gauge-invariant Quantities

The background equations (8) and the perturbation equations (32) are both written with respect to the same system of reference. Therefore, these two systems can be combined to describe the evolution of scalar perturbations. The system (8) describes the evolution of the quantities θ(0) = 3H, R(0), ε(0), ϑ(0) and n(0), whereas the system (32) describes the evolution of their perturbed counterparts θ(1), R(1), ε(1), ϑ(1) and n(1). The quantities ε, n and θ (10) are scalars. Let S represent one of the three scalars ε, n and θ, then a perturbation to these scalars transforms under a general inﬁnitesimal coordinate transformation (2) as

0 0 ˙ S(1)(t, x) → S(1) = S(1)(t, x) + ξ (t, x)S(0)(t), (41)

where S(0) and S(1) are the background and the perturbation, respectively, of one of these scalars. The ˆ 0 ˙ term S(1) := LξS(0) = ξ S(0) is the gauge mode, where Lξ is the Lie-derivative operator with respect to the infinitesimal four-vector ξµ. It can be easily verified by substitution that the complete set of gauge modes for the system of equations (32) is ˆ ˙ εˆ(1) = ψε˙(0), nˆ(1) = ψn˙ (0), θ(1) = ψθ(0), (42a) " # ∇˜ 2ψ ∇˜ 2ψ ϑˆ = − , Rˆ = 4H − 1 R ψ , (42b) (1) a2 (1) a2 2 (0) where ξ0 = ψ(x) in synchronous coordinates, see (4). The quantities (42) are mere coordinate artefacts, which have no physical meaning, since the gauge function ψ(x) is an arbitrary infinitesimal function. Since

11 the set (42) is a solution set of the system of linear equations (32), one can generate new, equivalent, solutions by adding the gauge modes (42) to the original solutions ε(1), n(1), θ(1), ϑ(1) and R(1). Therefore, the solutions of the system (32) have no physical significance. This is the well-known gauge problem of cosmology. A first step towards the solution of the gauge problem has already been taken in Section5, by rewriting the system of equations (11) into a substantially smaller system (32) that describes exclusively scalar perturbations. This has reduced the number of gauge-dependent quantities to five. The second step towards the solution of the gauge problem of cosmology can now be taken. Since ε(1), n(1) and θ(1) transform under the general infinitesimal transformation (2) according to (41), one can combine two independent scalars to eliminate the gauge function ξ0(t, x). With the three independent scalars (10), one can construct 3 2 = 3 different sets of three gauge-invariant quantities. In each of these sets exactly one gauge-invariant quantity vanishes. Therefore, the only non-trivial choice is ε˙ n˙ εgi := ε − (0) θ , ngi := n − (0) θ , (43a) (1) (1) ˙ (1) (1) (1) ˙ (1) θ(0) θ(0) θ˙ θgi := θ − (0) θ = 0. (43b) (1) (1) ˙ (1) θ(0) It follows from the transformation rule (41) that the quantities (43) are indeed invariant under the general infinitesimal transformation (2), i.e., they are gauge-invariant, hence the superscript ‘gi’. Since the evolution of flrw universes is determined by only three independent scalars (10), the quantities (43a) are the only non-trivial gauge-invariant quantities, i.e., these quantities are unique. From this fact one may draw gi phys the conclusion that the quantities (43a) are the true, physical density perturbations, i.e., ε(1) ≡ ε(1) and gi phys gi phys n(1) ≡ n(1) , and that the quantity (43b) θ(1) ≡ θ(1) = 0 is the true physical perturbation to the global expansion θ(0) of the universe. However, to be sure that this is indeed the case, one needs to study the system (32) together with the quantities (43) in the non-relativistic limit. This third and last step will be taken in the next section.

7 Non-relativistic Limit

The standard non-relativistic limit is deﬁned by four requirements, see, e.g., Carroll(2003), page 153:

1. The gravitational ﬁeld should be weak, i.e., can be considered as a perturbation of a ﬂat flrw universe.

2. The particles are moving slowly with respect to the speed of light.

3. The transformation (2) with (4) is limited to transformations for which the Newtonian Theory of Gravity in invariant.

4. The gravitational ﬁeld of a density perturbation should be static, i.e., it does not change with time.

Requirement 3, which is not used in the literature, is needed because one has to deal with gauge transformations (2) with (4) in an expanding universe. For cosmological perturbations the gravitational field is already weak. In order to meet the first requirement a flat, R(0) = 0, flrw universe is considered. In a flat flrw universe the expressions (43a) reduce to 1 εgi = − (2Hϑ + 1 R ), (44a) (1) κ (1) 2 (1) 1 gi n(0)(κε(1) + 2Hϑ(1) + 2 R(1)) n(1) = n(1) − , (44b) κε(0)(1 + w) where the background equations (8) have been used to eliminate all time-derivatives, the constraint equation (32a) to eliminate θ(1), and 3H = θ(0),

12 Using (21) and the fact that spatial covariant derivatives become in ﬂat three-space ordinary derivatives with respect to the spatial coordinates, the local perturbation to the spatial curvature, (17) and its gauge mode (42b), reduce for a ﬂat flrw universe to

4 4 ∇2φ ∇2ψ R = φ|k = − , Rˆ = 4H , (45) (1) c2 |k c2 a2 (1) a2 2 where ∇ is the usual Laplace operator. Substituting the expression for R(1) into the perturbation equations (32), one gets " # 1 ∇2φ 4πG ε˙ H(θ − ϑ ) = − + N εgi + (0) θ , (46a) (1) (1) 2 2 4 (1) ˙ (1) c a c θ(0) ∇2φ˙ 4πG = − N ε (1 + w)ϑ , (46b) a2 c2 (0) (1)

ε˙(1) = −3H(ε(1) + p(1)) − ε(0)(1 + w)θ(1), (46c) 2 ˙ 2 1 ∇ p(1) ϑ(1) = −H(2 − 3β )ϑ(1) − 2 , (46d) ε(0)(1 + w) a

n˙ (1) = −3Hn(1) − n(0)θ(1), (46e)

0 where (43a) has been used to eliminate ε(1) from the G(1)0 constraint equation (32a), and H :=a/a ˙ . With

R(0) = 0 the first requirement is satisfied. i Next, the second requirement is implemented. Since the spatial parts u(1) of the fluid four-velocity are gauge-dependent (42b), the second requirement must be defined by2 1 ui := U i → 0, (47) (1)phys c (1)phys i.e., the physical spatial parts of the fluid four-velocity are negligible with respect to the speed of light. With (47) the second requirement is satisfied. Next, the third requirement will be addressed. In the limit (47), the mean kinetic energy per particle 1 2 2 2 mhv i → 0 is very small compared to the rest energy mc per particle. This implies that the pressure 2 p → 0 is vanishingly small with respect to the rest energy density nmc . Substituting p = 0, i.e., p(0) = 0 and p(1) = 0, into the momentum conservation laws (11e) yields, using also the background equation (8c) with w := p(0)/ε(0) → 0, i i u˙ (1) = −2Hu(1). (48) i Since the components u(1)phys vanish in the non-relativistic limit (47), the general solutions of equations (48) are equal to the gauge modes, see (42b), 1 ∂ψ(x) uî (t, x) = − g˜ik(x) , (49) (1) a2(t) ∂xk where it is used that H :=a/a ˙ . As a result, in the limit (47) one is left with the gauge modes (49) only. i Since uˆ(1) = 0 are solutions of (48), one may, without losing any physical information, put the gauge modes i uˆ(1) equal to zero, i.e., i uˆ(1) = 0. (50) From (49) and (50) it now follows that ∂ψ/∂xk = 0, so that ψ = C is an arbitrary constant in the non- relativistic limit. Substituting ψ = C into (4) one finds that the relativistic transformation (2) between synchronous coordinates reduces in the limit (47) to the infinitesimal transformation

x0 → x00 = x0 − C, xi → x0i = xi − χi(x), (51)

2 Recall that u(1) = u(1)k is the irrotational part of the three-space ﬂuid velocity. The rotational part u(1)⊥ is not coupled to density perturbations and need, therefore, not be considered. See Section4.

13 where χi(x) are three arbitrary functions of the spatial coordinates. Thus, in the non-relativistic limit time and space transformations are decoupled: time coordinates may be shifted and spatial coordinates may be chosen arbitrarily. The residual gauge freedoms C and χi(x) in the non-relativistic limit express the fact that the Newtonian Theory of Gravity is invariant under the gauge transformation (51). This satisﬁes the third requirement. Next, the fourth and last requirement will be discussed. Using (47) and (50), on gets

k ϑ(1) := u(1)|k = 0. (52) Using (52) and p = 0, the system of Einstein equations and conservation laws (46) becomes in the non- relativistic limit: 4πG ∇2φ = N a2εgi , (53a) c2 (1) ∇2φ˙ = 0, (53b)

ε˙(1) = −3Hε(1) − ε(0)θ(1), (53c)

n˙ (1) = −3Hn(1) − n(0)θ(1), (53d)

0 The G(1)0 constraint equation (53a) can be found by observing that the time-derivative of the background 1 ˙ constraint equation, or Friedmann equation (8a), multiplied by 6 θ(1)/H, reads 4πG ε˙ Hθ = N (0) θ , (54) (1) 4 ˙ (1) c θ(0) where it is used that R(0) = 0 and θ(0) = 3H. Combining (46a), (52) and (54) yields equation (53a). ˆ The right-hand side of (53a) is clearly gauge-invariant. Since the gauge mode R(1) = 0,(45), R(1) is gauge-invariant, so that the left-hand side of (53a) is also gauge-invariant. Due to (52) there is no fluid flow so that density perturbations do not evolve. This, combined with (43b), implies the basic fact of the Newtonian Theory of Gravity, namely that the gravitational field is static, 2 gi as follows from the momentum constraint equation (53b). Consequently, a (t)ε(1)(t, x) in (53a) should be 2 gi 2 replaced by a (t0)ε(1)(x). Defining the potential ϕ(x) := φ(x)/a (t0), equations (53a) and (53b) imply

εgi (x) ∇2ϕ(x) = 4πG ρ (x), ρ (x) := (1) , (55) N (1) (1) c2 which is the Poisson equation of the Newtonian Theory of Gravity. With (55) the fourth requirement for the non-relativistic limit, i.e., a static gravitational ﬁeld, has been satisﬁed. gi Finally, expressions (44) will be checked. Using (52), expression (44a) reduces to ε(1) = −R(1)/2κ, which is, with (45) and (53b), equal to the Poisson equation (55). The background equations (8) are in the non-relativistic limit

2 3H = κε(0) + Λ, ε˙(0) = −3Hε(0), n˙ (0) = −3Hn(0). (56)

2 2 From equations (53c)–(53d) and (56) it follows that ε(1) = n(1)mc and ε(0) = n(0)mc , respectively, so that gi expression (44b) becomes, using that R(1) = −2κε(1), in a pressure-less ﬂuid

εgi (x) ngi (x) = (1) , ρ (x) = mngi (x), (57) (1) mc2 (1) (1) where the ﬁrst expression is a well-known result of the Special Theory of Relativity. Before a conclusion can be drawn, a few more remarks are made. From (52) and (53b) it follows that the perturbed expansion scalar (18) and its gauge mode (42a) are in the non-relativistic limit given by 1 θ = − ζ˙|k , θˆ = Cθ˙ , (58) (1) c2 |k (1) (0)

14 ˙ where (21) has been used. Using that θ(0) = 3H, it follows from (56) and (58) that θ(0) 6= 0 and θ(1) 6= 0, so that the quantities (43) are well-deﬁned. Consequently, it follows from (43a) that

gi gi ε(1) 6= ε(1), n(1) 6= n(1), (59) independent of the scale of a perturbation. Equations (53c) and (53d) are leftovers of the conservation laws (46c) and (46e) in the non-relativistic ˆ ˙ limit. They have only the gauge modes (42a) εˆ(1) = Cε˙(0), nˆ(1) = Cn˙ (0) and θ(1) = Cθ(0) as solutions and have, as a consequence, no physical signiﬁcance. These gauge modes never become zero, since the universe is not static in the non-relativistic limit, as follows from the Friedmann equation and conservation laws (56). Since equations (53c) and (53d) are decoupled from the physical equations (53a) and (53b) they are not part of the Newtonian Theory of Gravity and need, therefore, not be considered. Consequently, in the non-relativistic limit one is left with the Poisson equation (55) and the relation (57). This section will be closed with an important conclusion. It has been shown that the complete set of linearized Einstein equations and conservation laws for scalar perturbations (32), combined with the gauge- invariant quantities (43a) reduce in the non-relativistic limit to the Newtonian equations (55) and (57). gi gi This implies that the gauge-invariant quantities ε(1) and n(1) are equal to the true, physical perturbations phys phys gi ε(1) and n(1) , respectively, in the non-relativistic limit. It must, therefore, be concluded that ε(1) and gi phys phys n(1) are also equal to the true physical density perturbations ε(1) and n(1) in the General Theory of Relativity, i.e., ε˙ n˙ εgi := ε − (0) θ ≡ εphys, ngi := n − (0) θ ≡ nphys. (60) (1) (1) ˙ (1) (1) (1) (1) ˙ (1) (1) θ(0) θ(0) In view of the results in this section, (43b) can be written as

gi phys θ(1) ≡ θ(1) = 0, (61)

implying that, in the linear regime, the global expansion θ(0) = 3H is not aﬀected by a local density perturbation. With (60) and (61) the gauge problem of cosmology for closed, ﬂat and open flrw universes is solved.

8 Evolution Equations for the Density Contrast Functions

The evolution of the density perturbations (60) is completely determined by the background equations (8) and their perturbed counterparts (32). In principle, these two systems can be used to study the evolution of density perturbations in flrw universes. However, the system (32) is still too complicated, since it also phys phys admits the non-physical solutions (42). Therefore, evolution equations for ε(1) and n(1) are derived.

Firstly, it is observed that the gauge-dependent quantity θ(1) is not needed in the calculations, since phys the local perturbation to the global expansion, θ(1) (61), vanishes identically. Eliminating θ(1) from the differential equations (32b)–(32e) with the help of the algebraic constraint equation (32a) yields a system of four first-order ordinary differential equations h 1 i ε˙ + 3H(ε + p ) + ε (1 + w) ϑ + κε + 1 R = 0, (62a) (1) (1) (1) (0) (1) 2H (1) 2 (1) h 1 i n˙ + 3Hn + n ϑ + κε + 1 R = 0, (62b) (1) (1) (0) (1) 2H (1) 2 (1) ˜ 2 ˙ 2 1 ∇ p(1) ϑ(1) + H(2 − 3β )ϑ(1) + 2 = 0, (62c) ε(0)(1 + w) a R R˙ + 2HR − 2κε (1 + w)ϑ + (0) κε + 1 R = 0, (62d) (1) (1) (0) (1) 3H (1) 2 (1) for the four quantities ε(1), n(1), ϑ(1) and R(1).

15 Using the background equations (8) to eliminate all time-derivatives and the linearized constraint equation (32a) to eliminate θ(1), the density perturbations (60) become

1 phys ε(1)R(0) − 3ε(0)(1 + w)(2Hϑ(1) + 2 R(1)) ε(1) = , (63a) R(0) + 3κε(0)(1 + w) 1 phys 3n(0)(κε(1) + 2Hϑ(1) + 2 R(1)) n(1) = n(1) − . (63b) R(0) + 3κε(0)(1 + w)

The evolution of these density perturbations is completely determined by the background equations (8) and the set of first-order differential equations (62). In the study of the evolution of density perturbations, phys phys it is convenient not to use ε(1) and n(1) , but instead their corresponding contrast functions δε and δn, defined by phys phys ε(1) (t, x) n(1) (t, x) δε(t, x) := , δn(t, x) := . (64) ε(0)(t) n(0)(t)

The system of equations (62) for the four independent quantities ε(1), n(1), ϑ(1) and R(1) is now recast, using the procedure given in the appendix, in a new system of equations for the two independent contrast functions δε and δn. In this procedure it is explicitly assumed that p 6= 0, i.e., the pressure does not vanish identically. The case p → 0 has been considered in Section7 on the non-relativistic limit. The ﬁnal results are the evolution equations for the relative density perturbations (64) in flrw universes:

δ δ¨ + b δ˙ + b δ = b δ − ε , (65a) ε 1 ε 2 ε 3 n 1 + w 1 d δε 3Hn(0)pn δε δn − = δn − , (65b) c dt 1 + w ε(0)(1 + w) 1 + w where the coeﬃcients b1, b2 and b3 are given by

˙ 2 κε(0)(1 + w) β 2 1 2H(1 + 3β ) b1 = − 2 − H(2 + 6w + 3β ) + R(0) + , (66a) H β 3H R(0) + 3κε(0)(1 + w) ˙ 1 2 2 β κε(0)(1 + w) b2 = − 2 κε(0)(1 + w)(1 + 3w)H 1 − 3w + 6β (2 + 3w) + 6H w + β R(0) + 3κε(0)(1 + w) 2 2 ˜ 2 ! 1 H (1 + 6w)(1 + 3β ) 2 ∇ 1 − R(0) 2 w + − β 2 − 2 R(0) , (66b) R(0) + 3κε(0)(1 + w) a ( 2 " ˙ # ) ˜ 2 ! −18H 2pn β 2 n(0) ∇ 1 b3 = ε(0)pεn(1 + w) + + pn(pε − β ) + n(0)pnn + pn 2 − 2 R(0) . R(0) + 3κε(0)(1 + w) 3H β ε(0) a (66c)

In these expressions the partial derivatives of the pressure, pε and pn, are deﬁned by (12). The second-order partial derivatives are given by ∂2p ∂2p p := , p := . (67) nn ∂n2 εn ∂ε ∂n In the derivation of the coeﬃcients (66) it is used that the time-derivative of w is

w˙ = 3H(1 + w)(w − β2). (68)

2 This relation follows from the definitions w := p(0)/ε(0) and β :=p ˙(0)/ε˙(0) using only the energy conservation law (8c), and is, therefore, independent of the precise form of the equation of state. The system of equations (65) is equivalent to a system of three first-order differential equations, whereas the original system (62) is a fourth-order system. This difference is due to the fact that the gauge modes (42), which are solutions of the system (62), are completely removed from the solution set

16 of (65): one degree of freedom, namely the unknown gauge function ξ0(t, x) in (41) has disappeared al- ˙ together. This implies that one can impose initial values δε(t0, x), δε(t0, x) and δn(t0, x) which can, in principle, be obtained from observation and, subsequently, calculate the evolution of the true, physical, density contrast functions δε(t, x) and δn(t, x). The background equations (8) and the perturbation equations (65) constitute a set of equations which enables one to study the evolution of small fluctuations in the energy density δε and the particle number density δn in an open, flat and closed flrw universe with Λ 6≡ 0 and filled with a perfect fluid described by an equation of state for the pressure p = p(n, ε). This will be the subject of the companion article (Miedema, 2016).

9 Thermodynamics

In this section expressions for the true, physical pressure and temperature perturbations are derived. It is shown that density perturbations are adiabatic if and only if the particle number density does not contribute to the pressure.

9.1 Pressure and Temperature Perturbations

From the equation of state for the pressure p = p(n, ε), one has p˙(0) = pnn˙ (0) +pεε˙(0). Multiplying both sides ˙ of this expression by θ(1)/θ(0) and subtracting the result from p(1) given by (12), one gets, using also (43a), p˙ p − (0) θ = p nphys + p εphys. (69) (1) ˙ (1) n (1) ε (1) θ(0)

Hence, the quantity defined by p˙ pphys := p − (0) θ , (70) (1) (1) ˙ (1) θ(0) is the real pressure perturbation. Combining (69) and (70) and eliminating pε with the help of (33), one arrives at δ pphys = β2ε δ + n p δ − ε , (71) (1) (0) ε (0) n n 1 + w where also (64) has been used. The first term in this expression is the adiabatic part of the pressure perturbation and the second term is the diabatic part, as will be discussed in the next two subsections. From the equation of state (5) for the energy density ε = ε(n, T ) it follows that ∂ε ∂ε ˙ ∂ε ∂ε ε˙(0) = n˙ (0) + T(0), ε(1) = n(1) + T(1). (72) ∂n T ∂T n ∂n T ∂T n ˙ Multiplying ε˙(0) by θ(1)/θ(0) and subtracting the result from ε(1), one finds, using (60), " # ∂ε ∂ε T˙ εphys = nphys + T − (0) θ , (73) (1) (1) (1) ˙ (1) ∂n T ∂T n θ(0) implying that the quantity defined by

T˙ T phys := T − (0) θ , (74) (1) (1) ˙ (1) θ(0) is the real temperature perturbation. The expressions (70) and (74) are both of the form (60).

17 9.2 Diabatic Density Perturbations The combined First and Second Law of Thermodynamics is given by

dE = T dS − pdV + µdN, (75) where E, S and N are the energy, the entropy and the number of particles of a system with volume V and pressure p, and where µ, the thermal—or chemical—potential, is the energy needed to add one particle to the system. In terms of the particle number density n = N/V , the energy per particle E/N = ε/n and the entropy per particle s = S/N the law (75) can be rewritten as

ε N d N = T d(sN) − pd + µdN, (76) n n where ε is the energy density. The system is extensive, i.e., S(λE, λV, λN) = λS(E,V,N), implying that the entropy of the gas is S = (E + pV − µN)/T . Dividing this relation by N one gets the so-called Euler relation ε + p µ = − T s. (77) n Eliminating µ in (76) with the help of (77), one ﬁnds that the combined First and Second Law of Thermo- dynamics (75) can be cast in a form without µ and N, i.e., ε 1 T ds = d + pd . (78) n n

From the background equations (8) and the thermodynamic law (78) it follows that s˙(0) = 0, implying that gi the expansion of the universe takes place without generating entropy. Using (41) one ﬁnds that s(1) = s(1) is gi phys automatically gauge-invariant. Consequently, s(1) ≡ s(1) is the true, physical perturbation to the entropy per particle. The thermodynamic relation (78) can, using (64) and w := p(0)/ε(0), be rewritten as phys ε(0)(1 + w) δε T(0)s(1) = − δn − . (79) n(0) 1 + w

From (79) it follows that the right-hand side of equation (65a) is related to local perturbations in the entropy, and equation (65b) can be considered as an evolution equation for entropy perturbations. For phys T(0)s(1) 6= 0 density perturbations are diabatic, i.e., they exchange heat with their environment.

9.3 Adiabatic Density Perturbations In the limiting case that the contribution of the particle number density to the pressure is negligible, i.e., pn ≈ 0, the coeﬃcient b3,(66c), vanishes, so that the evolution equation (65a) becomes homogeneous. In this case equation (65a) describes a closed system which does not exchange heat with its environment. In other words, the system is adiabatic and evolves only under its own gravity. phys For pn ≈ 0, the right-hand side of the entropy evolution equation (65b) vanishes. Since T(0)s(1) = 0 for adiabatic perturbations, one ﬁnds from (79) that the solution of (65b) is

δ (t, x) δ (t, x) − ε = 0. (80) n 1 + w(t)

This relation implies that perturbations in the particle number density, δn, are through gravitation coupled to perturbations in the (total) energy density, δε, independent of the nature of the particles. phys 2 The pressure perturbation (71) becomes p(1) = β ε(0)δε, i.e., only the adiabatic part of the pressure perturbation survives. Therefore, in a ﬂuid described by a barotropic equation of state, p ≈ p(ε), density perturbations evolve adiabatically. In all other cases where p = p(n, ε), local density perturbations evolve diabatically.

18 10 Conclusion

In this article a new approach to the theory of cosmological perturbations in flrw universes has been developed. By using the decomposition theorems, Section4, to the fullest extent, the set of linearized Einstein equations and conservation laws can for scalar perturbations be recast in the form (32) which is the perturbed counterpart of the background Einstein equations and conservation laws (8). From the systems of equations (8) and (32) it follows that only three scalars ε, n and θ and their perturbations play a role in the evolution of density perturbations. Using the transformation rule (41) one can construct one and only one non-trivial set of gauge-invariant quantities (43). Since the General Theory of Relativity phys is a mathematically consistent theory, there exists one and only one quantity ε(1) for the perturbation phys to the energy density and one and only one quantity n(1) for the perturbation to the particle number gi gi density. Therefore, the unique gauge-invariant quantities ε(1) and n(1) found in Section6 must be equal to phys phys the true physical density perturbations ε(1) and n(1) , respectively. In Section7 on the non-relativistic limit it is shown that this is indeed the case. In fact, it is shown that the system of equations (32) and the definitions (43a) reduce in the non-relativistic to the time-independent Poisson equation (55) and the relation (57). The gauge transformation of the General Theory of Relativity reduces in the non-relativistic limit to the gauge transformation of the Newtonian Theory of Gravity. That is why in the General Theory of Relativity a coordinate system cannot be uniquely fixed, since the Newtonian Theory of Gravity is invariant under the residual gauge transformation (51). phys phys In Section8 evolution equations for density perturbations ε(1) and n(1) in closed, flat and open flrw universes are derived. What is done, essentially, is to rewrite the linearized Einstein equations and conservation laws for the quantities ε(1) and n(1) in terms of the measurable energy and particle phys phys number densities ε(1) and n(1) , just as in electromagnetism the Coulomb and Ampère laws for the scalar potential Φ and vector potential A can be rewritten in the form of the Maxwell equations for the measurable electric field E and magnetic field B. The treatment of cosmological perturbations presented in this article has the following properties.

Firstly, in equations (32) all metric tensor perturbations are contained in only three quantities: R(1),(17),

θ(1) and ϑ(1),(18). This eliminates completely the unwanted gradient terms (Malik and Matravers, 2013) in the resulting equations (65). Furthermore, the quantities (60) do not contain spatial derivatives (Ellis ˙ and Bruni, 1989; Ellis et al., 1989; Ellis and van Elst, 1998). Secondly, due to the quotients ε˙(0)/θ(0) and ˙ phys phys n˙ (0)/θ(0) of time derivatives, the quantities ε(1) and n(1) ,(60), are independent of the definition of time, so that their evolution is only determined by their propagation equations. Finally, from (61) it follows that, in the linear regime, the global expansion θ(0) = 3H is not affected by a local density perturbation. This is in accordance with the results found by Green and Wald(2014). In a fluid described by a barotropic equation of state, p ≈ p(ε), the right-hand side of the evolution equation (65a) vanishes and equation (65b) reduces to an identity, so that density perturbations evolve adiabatically. In all other cases where p = p(n, ε), local density perturbations evolve diabatically. In this article an equation of state for the pressure p(n, ε) in a perfect fluid is used. However, the algorithm in the appendix can be easily adapted to more general cases. The equations (65) have been checked using the computer algebra system Maxima(2020), as follows. phys phys Substituting the contrast functions (64) into equations (65), where ε(1) and n(1) are given by (63), and subsequently eliminating the time-derivatives of ε(0), n(0), H, R(0) and ε(1), n(1), ϑ(1), R(1) with the help of equations (8) and (62), respectively, yields two identities for each of the two equations (65). In a companion article (Miedema, 2016) the evolution equations (65) will be solved. Since the gauge problem of cosmology is solved, the physical consequences of the new approach to cosmological perturbations stand out clearly.

19 A Derivation of the Evolution Equations using Computer Algebra

In this appendix the perturbation equations (65) of the main text are derived from the basic perturbation equations (62) and the deﬁnitions (64). This is done by ﬁrst deriving the evolution equations for the phys phys gauge-invariant quantities ε(1) and n(1) (63): phys phys phys phys n(0) phys ε¨(1) + a1ε˙(1) + a2ε(1) = a3 n(1) − ε(1) , (A.1a) ε(0)(1 + w) 1 d phys n(0) phys n(0)pn phys n(0) phys n(1) − ε(1) = −3H 1 − n(1) − ε(1) . (A.1b) c dt ε(0)(1 + w) ε(0)(1 + w) ε(0)(1 + w)

The coeﬃcients a1, a2 and a3 occurring in equation (A.1a) are given by ˙ 2 κε(0)(1 + w) β 2 1 2H(1 + 3β ) a1 = − 2 + H(4 − 3β ) + R(0) + , (A.2a) H β 3H R(0) + 3κε(0)(1 + w)  ˙  2 2 β ˙ 5H (1 + 3β ) − 2H ˜ 2 ! β 2 2 1 β  2 ∇ 1 a2 = κε(0)(1 + w) − 4H + 2H (2 − 3β ) + R(0)  +  − β 2 − 2 R(0) , β 2 R(0) + 3κε(0)(1 + w)  a

(A.2b) ( 2 " ˙ # ) ˜ 2 ! −18H 2pn β 2 ∇ 1 a3 = ε(0)pεn(1 + w) + + pn(pε − β ) + n(0)pnn + pn 2 − 2 R(0) . R(0) + 3κε(0)(1 + w) 3H β a (A.2c)

In calculating the coeﬃcients a1, a2 and a3,(A.2), it is used that the time derivative of the quotient 2 w := p(0)/ε(0) is given by (68). Moreover, it is convenient not to expand the quantity β :=p ˙(0)/ε˙(0) since this considerably complicates the expressions for the coeﬃcients a1, a2 and a3.

A.1 Derivation of the Evolution Equation for the Energy Density Perturbation In order to derive equation (A.1a), the system (62) and expression (63a) is rewritten, using (12), in the form

ε˙(1) + α11ε(1) + α12n(1) + α13ϑ(1) + α14R(1) = 0, (A.3a)

n˙ (1) + α21ε(1) + α22n(1) + α23ϑ(1) + α24R(1) = 0, (A.3b) ˙ ϑ(1) + α31ε(1) + α32n(1) + α33ϑ(1) + α34R(1) = 0, (A.3c) ˙ R(1) + α41ε(1) + α42n(1) + α43ϑ(1) + α44R(1) = 0, (A.3d) phys ε(1) + α51ε(1) + α52n(1) + α53ϑ(1) + α54R(1) = 0, (A.3e) where the coeﬃcients αij are given in Table1.

Step 1 First the quantity R(1) is eliminated from equations (A.3). Diﬀerentiating equation (A.3e) with ˙ ˙ respect to time and eliminating the time derivatives ε˙(1), n˙ (1), ϑ(1) and R(1) with the help of equations (A.3a)–(A.3d), one arrives at the equation

phys ε˙(1) + p1ε(1) + p2n(1) + p3ϑ(1) + p4R(1) = 0, (A.4) where the coeﬃcients p1, . . . , p4 are given by

pi =α ˙ 5i − α51α1i − α52α2i − α53α3i − α54α4i. (A.5) From equation (A.4) it follows that

1 phys p1 p2 p3 R(1) = − ε˙(1) − ε(1) − n(1) − ϑ(1). (A.6) p4 p4 p4 p4

20 Table 1: The coeﬃcients αij ﬁguring in the equations (A.3).

κε (1 + w) ε (1 + w) 3H(1 + p ) + (0) 3Hp ε (1 + w) (0) ε 2H n (0) 4H κn n (0) 3H n (0) 2H (0) 4H

˜ 2 ˜ 2 pε ∇ pn ∇ 2 2 2 H(2 − 3β ) 0 ε(0)(1 + w) a ε(0)(1 + w) a

κR R (0) 0 −2κε (1 + w) 2H + (0) 3H (0) 6H

3 −R 6ε H(1 + w) ε(0)(1 + w) (0) 0 (0) 2 R(0) + 3κε(0)(1 + w) R(0) + 3κε(0)(1 + w) R(0) + 3κε(0)(1 + w)

phys In this way the quantity R(1) has been expressed as a linear combination of the quantities ε˙(1) , ε(1), n(1) and ϑ(1). Upon replacing R(1) in equations (A.3) by the right-hand side of (A.6), one arrives at the system of equations

phys ε˙(1) + q1ε˙(1) + γ11ε(1) + γ12n(1) + γ13ϑ(1) = 0, (A.7a) phys n˙ (1) + q2ε˙(1) + γ21ε(1) + γ22n(1) + γ23ϑ(1) = 0, (A.7b) ˙ phys ϑ(1) + q3ε˙(1) + γ31ε(1) + γ32n(1) + γ33ϑ(1) = 0, (A.7c) ˙ phys R(1) + q4ε˙(1) + γ41ε(1) + γ42n(1) + γ43ϑ(1) = 0, (A.7d) phys phys ε(1) + q5ε˙(1) + γ51ε(1) + γ52n(1) + γ53ϑ(1) = 0, (A.7e) where the coeﬃcients qi and γij are given by

αi4 qi = − , γij = αij + qipj. (A.8) p4

It has now been achieved that the quantity R(1) occurs explicitly only in equation (A.7d), whereas R(1) occurs implicitly in the remaining equations. Therefore, equation (A.7d) is not needed anymore. Equations

(A.7a)–(A.7c) and (A.7e) are four ordinary diﬀerential equations for the four unknown quantities ε(1), n(1), phys ϑ(1) and ε(1) .

Step 2 In the same way as in Step 1, the explicit occurrence of the quantity ϑ(1) is eliminated from the system of equations (A.7). Diﬀerentiating equation (A.7e) with respect to time and eliminating the time ˙ derivatives ε˙(1), n˙ (1) and ϑ(1) with the help of equations (A.7a)–(A.7c), one arrives at

phys phys q5ε¨(1) + rε˙(1) + s1ε(1) + s2n(1) + s3ϑ(1) = 0, (A.9) where the coeﬃcients si and r are given by

si =γ ˙ 5i − γ51γ1i − γ52γ2i − γ53γ3i, (A.10a)

r = 1 +q ˙5 − γ51q1 − γ52q2 − γ53q3. (A.10b) From equation (A.9) it follows that

q5 phys r phys s1 s2 ϑ(1) = − ε¨(1) − ε˙(1) − ε(1) − n(1). (A.11) s3 s3 s3 s3

21 phys phys In this way the quantity ϑ(1) is expressed as a linear combination of the quantities ε¨(1) , ε˙(1) , ε(1) and n(1). Upon replacing ϑ(1) in equations (A.7) by the right-hand side of (A.11), one arrives at the system of equations q5 phys r phys s1 s2 ε˙(1) − γ13 ε¨(1) + q1 − γ13 ε˙(1) + γ11 − γ13 ε(1) + γ12 − γ13 n(1) = 0, (A.12a) s3 s3 s3 s3 q5 phys r phys s1 s2 n˙ (1) − γ23 ε¨(1) + q2 − γ23 ε˙(1) + γ21 − γ23 ε(1) + γ22 − γ23 n(1) = 0, (A.12b) s3 s3 s3 s3 ˙ q5 phys r phys s1 s2 ϑ(1) − γ33 ε¨(1) + q3 − γ33 ε˙(1) + γ31 − γ33 ε(1) + γ32 − γ33 n(1) = 0, (A.12c) s3 s3 s3 s3 ˙ q5 phys r phys s1 s2 R(1) − γ43 ε¨(1) + q4 − γ43 ε˙(1) + γ41 − γ43 ε(1) + γ42 − γ43 n(1) = 0, (A.12d) s3 s3 s3 s3 phys q5 phys r phys s1 s2 ε(1) − γ53 ε¨(1) + q5 − γ53 ε˙(1) + γ51 − γ53 ε(1) + γ52 − γ53 n(1) = 0. (A.12e) s3 s3 s3 s3

It has now been achieved that the quantities ϑ(1) and R(1) occur explicitly only in equations (A.12c) and (A.12d), whereas they occur implicitly in the remaining equations. Therefore, equations (A.12c) and (A.12d) are not needed anymore. Equations (A.12a), (A.12b) and (A.12e) are three ordinary diﬀerential phys equations for the three unknown quantities ε(1), n(1) and ε(1) .

Step 3 At first sight, the next steps would be to eliminate, successively, the quantities ε(1) and n(1) from equation (A.12e) with the help of equations (A.12a) and (A.12b). One would then end up with a phys fourth-order differential equation for the unknown quantity ε(1) . This, however, is impossible, since the gauge-dependent quantities ε(1) and n(1) do not occur explicitly in equation (A.12e), as is now shown. Firstly, it is observed that equation (A.12e) can be rewritten as phys phys phys γ51s3 − γ53s1 ε¨(1) + a1ε˙(1) + a2ε(1) = a3 n(1) + ε(1) , (A.13) γ52s3 − γ53s2 where the coefficients a1, a2 and a3 are given by

s3 r s3 γ52s3 s2 a1 = − + , a2 = − , a3 = − . (A.14) γ53 q5 γ53q5 γ53q5 q5 These are the coeﬃcients (A.2). Secondly, one ﬁnds γ s − γ s n 51 3 53 1 = − (0) . (A.15) γ52s3 − γ53s2 ε(0)(1 + w) Finally, using the expressions (60) and the conservation laws (8c) and (8e), one gets

n(0) phys n(0) phys n(1) − ε(1) = n(1) − ε(1) . (A.16) ε(0)(1 + w) ε(0)(1 + w)

Consequently, the right-hand side of (A.13) does not explicitly contain the gauge-dependent quantities ε(1) and n(1). With the help of expression (A.16) one can rewrite equation (A.13) in the form (A.1a). The derivation of the coeﬃcients (A.2) from (A.14) and the proof of the equality (A.15) is straightfor- ward, but extremely complicated. The computer algebra system Maxima(2020) has been used to perform this algebraic task.

A.2 Derivation of the Evolution Equation for the Entropy Perturbation The basic set of equations (62) from which the evolution equations are derived is of fourth-order. From phys this system a second-order equation (A.1a) for ε(1) has been extracted. Therefore, the remaining system phys from which an evolution equation for n(1) can be derived is at most of second order. Since gauge-invariant phys phys 0 quantities ε(1) and n(1) have been used, one degree of freedom, namely the gauge function ξ (t, x) in (41)

22 phys has disappeared. As a consequence, only a ﬁrst-order evolution equation for n(1) can be derived. Instead phys of deriving an equation for n(1) , an evolution equation (A.1b) for the entropy perturbation, which contains phys n(1) , is derived. From the combined First and Second Law of Thermodynamics (78) it follows that ε(0)(1 + w) n(0) T(0)s(1) = − 2 n(1) − ε(1) , (A.17) n(0) ε(0)(1 + w)

phys where the right-hand side is gauge-invariant by virtue of (A.16), so that s(1) = s(1) is gauge-invariant, in accordance with the remark below (78). Differentiating the term between square brackets in (A.17) with respect to time and using the background equations (8c) and (8e), the first-order equations (62a) and (62b) 2 and the definitions w := p(0)/ε(0) and β :=p ˙(0)/ε˙(0), one finds 1 d n(0) n(0)pn n(0) n(1) − ε(1) = −3H 1 − n(1) − ε(1) , (A.18) c dt ε(0)(1 + w) ε(0)(1 + w) ε(0)(1 + w) where Maxima(2020) has been used to perform the algebraic task. By virtue of (A.16), one may in this phys phys equation replace n(1) and ε(1) by n(1) and ε(1) , respectively, thus obtaining equation (A.1b).

A.3 Evolution Equations for the Contrast Functions First the entropy equation (65b) is derived. From the deﬁnitions (64) it follows that phys n(0) phys δε n(1) − ε(1) = n(0) δn − . (A.19) ε(0)(1 + w) 1 + w Substituting this expression into equation (A.18) one ﬁnds, using also (A.16),

1 d δε n(0)pn δε n(0) δn − = −3H 1 − n(0) δn − . (A.20) c dt 1 + w ε(0)(1 + w) 1 + w Using equation (8e) one arrives at equation (65b) of the main text. Finally, equation (65a) is derived. Upon substituting the expressions

phys phys ˙ phys ˙ ¨ ε(1) = ε(0)δε, ε˙(1) =ε ˙(0)δε + ε(0)δε, ε¨(1) =ε ¨(0)δε + 2ε ˙(0)δε + ε(0)δε, (A.21)

into equation (A.1a), and dividing by ε(0), one ﬁnds

ε˙(0) ε¨(0) ε˙(0) n(0) b1 = 2 + a1, b2 = + a1 + a2, b3 = a3 , (A.22) ε(0) ε(0) ε(0) ε(0) where also (A.19) has been used. Using Maxima(2020), one arrives at the coeﬃcients (66) of the main text.

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